In the affine space \(\mathbb{F}_q^n\) over the finite field of order \(q\), a point set \(S\) is said to be \((d,k,r)\)-evasive if the intersection between \(S\) and any variety, of dimension \(k\) and degree at most \(d\), has cardinality less than \(r\). As \(q\) tends to infinity, the size of a \((d,k,r)\)-evasive set in \(\mathbb{F}_q^n\) is at most \(O(q^{n-k})\) by a simple averaging argument. We exhibit the existence of such evasive sets of sizes at least \(\Omega(q^{n-k})\) for much smaller values of \(r\) than previously known constructions, and establish an enumerative upper bound \(2^{O(q^{n-k})}\) for the total number of such evasive sets. The existence result is based on our study of twisted varieties. In the projective space \(\mathbb{P}^n\) over an algebraically closed field, a variety \(V\) is said to be \(d\)-twisted if the intersection between \(V\) and any variety of dimension \(n-\dim(V)\) and degree at most \(d\) has dimension zero. We prove an upper bound on the smallest possible degree of twisted varieties which is best possible in a mild sense. The enumeration result includes a new technique for the container method which we believe is of independent interest. To illustrate the potential of this technique, we give a simpler proof of a result by Chen–Liu–Nie–Zeng that characterizes the maximum size of a collinear-triple-free subset in a random sampling of \(\mathbb{F}_q^2\) up to polylogarithmic factors.

Abstract. The [Formula: see text]-expansion of a [Formula: see text]-uniform hypergraph [Formula: see text], denoted by [Formula: see text], is an [Formula: see text]-uniform hypergraph obtained by enlarging each [Formula: see text]-edge of [Formula: see text] with a set of [Formula: see text] vertices of degree one. The random Turán number [Formula: see text] is the maximum number of edges in an [Formula: see text]-free subgraph of [Formula: see text], where [Formula: see text] is the Erdős–Rényi random [Formula: see text]-graph with parameter [Formula: see text]. In this paper, we prove an upper bound for [Formula: see text] when [Formula: see text] belongs to a large family of [Formula: see text]-partite [Formula: see text]-graphs: the [Formula: see text]-expansion of spanning subgraphs of tight trees. This upper bound is essentially tight for at least the following two families of hypergraphs: 1. Our upper bounds are essentially tight for expansions of [Formula: see text], the complete [Formula: see text]-graph on [Formula: see text] vertices. The proof of the lower bound makes use of a recent construction of Gowers and Janzer generalizing the famous Ruzsa–Szemerédi construction. In particular, when [Formula: see text], this answers a question of the current author, Spiro, and Verstraëte concerning the random Turán number of linear triangle. 2. Let [Formula: see text] be a tight tree such that the intersection of all edges of [Formula: see text] is empty. Simple construction shows that the upper bounds we have for expansions of [Formula: see text] are essentially tight. The main technical contribution of this paper is a new way to obtain balanced supersaturation results for expansions of hypergraphs: we combine two ideas, one of Mubayi and Yepremyan and another of Balogh, Narayanan, and Skokan, via codegree dichotomy. We note that neither of these two ideas alone would be enough to recover results in this paper.

In this paper, we consider an extension of cycle-complete graph Ramsey numbers to Berge cycles in hypergraphs: for $k \geq 2$, a nontrivial Berge $k$-cycle is a family of sets $e_1,e_2,\dots,e_k$ such that $e_1 \cap e_2, e_2 \cap e_3,\dots,e_k \cap e_1$ has a system of distinct representatives and $e_1 \cap e_2 \cap \dots \cap e_k = \emptyset$. In the case that all the sets $e_i$ have size three, let $\mathcal{B}_k$ denote the family of all nontrivial Berge $k$-cycles. The Ramsey numbers $R(t,\mathcal{B}_k)$ denote the minimum $n$ such that every $n$-vertex 3-uniform hypergraph contains either a nontrivial Berge $k$-cycle or an independent set of size $t$. We prove $R(t, \mathcal{B}_{2k}) \leq t^{1 + \frac{1}{2k-1} + \frac{2}{\sqrt{\log t}}}$, and moreover, we show that if a conjecture of Erdös and Simonovits [ Combinatorica, 2 (1982), pp. 275--288] on girth in graphs is true, then this is tight up to a factor $t^{o(1)}$ as $t \rightarrow \infty$.